The Riemann Hypothesis Applies Not Only to Prime Numbers

Abstract

The analysis of a series of natural numbers 0, 1, 2, 3, 4, ..., 1024 is given after expansion in the binary number system. Particular cases of this series are composite, even and odd numbers, and the latter are divided into prime numbers and composite odd numbers. Comparison of these series is carried out according to the general formula for the distribution of binary numbers for all 9 digits of the binary number system. It is shown that the shift of the oscillation of a binary number is different for varieties of natural numbers. It is proved that the real root of the Riemann zeta function 1/2 exists for any series of numbers obtained from the series of natural numbers. In the limit, with an increase in the power of the series, a sinusoid is subtracted from the real root (for series of odd, prime and odd composite numbers) and a cosine function (for series of natural, composite and integer numbers), the amplitude of which is also equal to 1/2, and the half-period of the trigonometric function are two numbers: 1 —for series of natural and composite natural numbers; 2—for series of odd natural, prime, composite odd and even numbers. Moreover, under the sine and cosine functions, the varieties of series of natural numbers are located on the Riemannian critical lines in the following way: a) on the zero vertical of the binary expansion of series of natural and composite numbers, the value π; b) on the first vertical of the binary expansion of series of odd, prime, composite odd and integer natural numbers π/2.